Rapidly Converging Bounds for the Ground-State Energy of Hydrogenic Atoms in Superstrong Magnetic Fields

Abstract

The calculation of rapidly converging lower and upper bounds to the ground-state energy, E g , of hydrogenic atoms in superstrong magnetic fields (B^\0 9 G) has been an important theoretical problem for the past twenty-five years. Much effort has gone into reconciling the many different estimates for E g predicted by an assortment of techniques. On the basis of recently developed eigenvalue moment methods, a precise solution involving rapidly converging bounds to E g , for arbitrary superstrong magnetic field strengths, is now possible.

Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure

“…The most famous of these was the notoriously difficult, strong coupling, singular perturbation problem corresponding to determining the ground state binding energy of the quadratic Zeeman effect for super-strong magnetic fields (of the type encountered in neutron stars). Handy et al [24,25] were able to extend the EMM analysis to such problems confirming, through tight bounds, the results of Le Guillou and Zinn-Justin [26] derived through more sophisticated, analytical methods.…”

“…An alternative approach (for multidimensional systems) is to use LUdecomposition methods on the (multidimensional) Hankel moment matrix (for the weight), in order to generate the same orthogonal polynomials [39]. One interesting application of this is to use EMM [20,24,25] to determine the power moments of the physical (positive) bosonic ground state (through tight bounds), for a given system. These can then be incorporated within OPPQ to generate the discrete state energies for all of the desired discrete states.…”

“…Thus, one could use the (unknown) bosonic ground state (which must be positive), provided its power moments can be generated to high accuracy, enabling the generation of its corresponding orthonormal polynomials. In principle, the Eigenvalue Moment Method (EMM) could be used for such cases (generating high precision values for the power moments of the ground state, as well as the energy, through converging bounds) [20,24,25].…”

Extension of a spectral bounding method to the PT-invariant states of the −(iX)Nnon-Hermitian potential

“…Because of this we can recast the quantization problem into a Moment Prob lem [22] formulation and exploit the applicability of the Eigenvalue Moment Method (EMM), as originally developed by Handy and Bessis [23], and its subsequent, linear programming based, reformulation by Handy et al [24]. The remarkable feature of the EMM approach is that the physical eigenenergies are obtained through a conver gent sequence of lower and upper bound approximants.…”

“…Handy et al [24] applied the EMM procedure to this notoriously difficult problem, generating tight bounds on the binding energy of one electron systems in intense magnetic fields (i.e., neutron stars, etc.) of the order of iO~Gauss and higher.…”